Generalization of conjugacy classes for all automorphisms

Question

In group $G$, the conjugacy class $\text{Cl}(a)$ of an element $a\in G$ is defined as $\{gag^{-1}:g\in G\}$. This can be rewritten as $\{\varphi(a):\varphi\in\text{Inn}(G)\}$, where $\text{Inn}(G)$ is the inner automorphism group of $G$, that is, the group of conjugation automorphisms.

But what if we do this for any automorphism? That is, define the automorphism class $\text{Al}(a)=\{\varphi(a):\varphi\in\text{Aut}(G)\}$. Just like usual conjugacy classes, automorphism classes partition $G$, and if two elements are in the same automorphism class, then they are indistinguishable by the group structure alone.

Moreover, an automorphism class must be a disjoint union of conjugacy classes. Thus, a few questions:

1. Must the conjugacy classes that comprise the automorphism class have the same cardinality?
2. If yes, must there be the same number of conjugacy classes for every automorphism class?
3. If yes, is this number $|Out(G)|$?

EDIT: The answer to the second question is NO. So I change the third question to:

3. If $Out(G)$ is finite, does the number of conjugacy classes always divide $|Out(G)|$?

Answer 1

1. Yes. Suppose that $C$ is a conjugacy class of $x \in G$. Then for $\varphi \in Aut(G)$, the image $\varphi(C)$ is the conjugacy class of $\varphi(x)$. Here $|\varphi(C)| = |C|$ since $\varphi$ is a bijection. The automorphism class of $C$ is a disjoint union of of some $\varphi(C)$'s.
2. No, for example the identity is fixed by every automorphism, so the automorphism class is $\{1\}$. But there are groups where an automorphism class contains more than one conjugacy class.

For an instructive exercise, compute the automorphism classes of $G = D_8$, the dihedral group of order $8$.